If the word ‘UNIVERSITY’ is arranged randomly, then the probability that both ‘I’ are not together, is

Options

A. `3/5`
B. `2/5`
C. `4/5`
D. `3/5`

Answer

To find the probability that both 'I's in the word "UNIVERSITY" are not together, we can follow these steps:

### Step 1: Calculate the total arrangements of the word "UNIVERSITY".

The word "UNIVERSITY" consists of 10 letters where the letter 'I' appears twice. The formula to calculate the total arrangements of letters when some letters are repeated is given by:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]

where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

For "UNIVERSITY":
- Total letters, \( n = 10 \)
- The letter 'I' appears 2 times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{10!}{2!} \]

### Step 2: Calculate the arrangements where both 'I's are together.

To find the arrangements where both 'I's are together, we can treat the two 'I's as a single entity or block. Therefore, we can think of the block 'II' as one letter.

Now, the letters we have are: U, N, II, V, E, R, S, T, Y (which gives us 9 letters).

The arrangements of these 9 letters are:

\[ \text{Arrangements with 'I's together} = 9! \]

### Step 3: Calculate the probability that both 'I's are together.

The probability that both 'I's are together is given by the ratio of the arrangements where 'I's are together to the total arrangements:

\[ P(\text{both 'I's together}) = \frac{\text{Arrangements with 'I's together}}{\text{Total arrangements}} = \frac{9!}{\frac{10!}{2!}} \]

### Step 4: Calculate the probability that both 'I's are not together.

The probability that both 'I's are not together is the complement of the probability that both 'I's are together:

\[ P(\text{both 'I's not together}) = 1 - P(\text{both 'I's together}) \]

Substituting the values we calculated:

\[ P(\text{both 'I's not together}) = 1 - \frac{9!}{\frac{10!}{2!}} = 1 - \frac{9! \times 2!}{10!} \]

### Step 5: Simplify the expression.

We know that \( 10! = 10 \times 9! \), thus:

\[ P(\text{both 'I's not together}) = 1 - \frac{2}{10} = 1 - 0.2 = 0.8 \]

### Final Answer:

The probability that both 'I's are not together is \( 0.8 \).

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